3. CONCLUSIONS
In this article, we have presented a dual-band planar GNSS
antenna with a wideband circular polarization. The different stand-
ards covered are the GPS, GLONASS, and Galileo. The return
loss bandwidth is around 22% in the L
1
band and 14% in the other
band L
2
. The AR bandwidth is 20% in the first band and 11.3% in
the second one. The radiation pattern shows that the antenna has 6
and 8.4 dBi in the normal direction at the central frequencies 1.23
and 1.6 GHz, respectively. The total dimensions of the antenna are
170 190 mm
2
with a thickness of 7.651 mm.
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V
C
2012 Wiley Periodicals, Inc.
A SIMPLE AND EFFICIENT FINITE
DIFFERENCE FREQUENCY DOMAIN
METHOD FOR THE ANALYSIS AND
DESIGN OF LEAKY-WAVE ANTENNAS
Feng Xu
School of Electronic Science and Engineering, Nanjing University of
Posts and Telecommunications, Nanjing 210003, China;
Corresponding author: feng.xu@njupt.edu.cn
Received 25 March 2012
ABSTRACT: The finite difference frequency domain method for the
analysis of periodic structures will lead to a large-scale nonsymmetrical
generalized eigenvalue problem, especially when the open periodic
structures are analyzed. In this article, the method is greatly simplified,
because the large-scale sparse matrix technique and the shift-and-invert
Arnoldi technique, which must be applied to significantly reduce the
memory need and increase simulation speed, are implemented by means
of some existed codes, for example, matlab codes. Besides, the
difference equations of this method are much simpler than those of the
previous method, because the longitudinal field components do not have
to be eliminated to approximately transform the generalized eigenvalue
problem to a standard eigenvalue problem. As a result, the method
proposed here is quite simple and efficient for the analysis of leaky-
wave antenna, because a leaky-wave antenna can be regarded as direct
geometrical evolution from a guided-wave structure that permits to leak
energy along its longitudinal direction.
V
C
2012 Wiley Periodicals, Inc.
Microwave Opt Technol Lett 54:2814–2817, 2012; View this article
online at wileyonlinelibrary.com. DOI 10.1002/mop.27216
Key words: Arnoldi techniques; finite difference frequency domain
method; large nonsymmetrical generalized eigenvalue problem;
leaky-wave antenna; periodic guided-wave structure
1. INTRODUCTION
The determination of dispersion characteristics of a periodic
guided-wave structure is a very important subject in practical
RF engineering design. In slow-wave structures, backward-wave
oscillators, corrugated antennas, and leaky-wave antennas, peri-
odic structures have been measured and investigated by many
researchers due to their importance [1–5]. Accurate prediction
of the propagation behavior of periodic structures is an essential
step in the successful design of these systems.
Although a lot of commercial software can be used to ana-
lyze the periodic structures, the complex propagation constants
cannot be obtained directly, because there is no accurate single
periodic structure model. Usually, twice simulations for the
same periodic structure but with different lengths should be
done. After that, the accurate propagation constants can be
obtained from these parameters using the so-called numerical
calibration technique. Some commercial software provides sin-
gle periodic structure model, for example, the eigenvalue solu-
tion (used for resonant cavities) of HFSS. Using this solution
and combining the equivalent resonant cavity model [6, 7], in
some cases, one can accurately extract the propagation constant.
However, when the attenuation constant is large, for example,
the case of the leaky-wave antennas, this solution is not accu-
rate. Especially, when the phase between two periodic planes is
close to 180
, the solution will crash down.
On the other side, a variety of numerical techniques have
been used to analyze periodic guided-wave structures. Among
them, the finite difference frequency domain (FDFD) method is
one of the most versatile techniques for solving partial differen-
tial equations, and it has been used to analyze modal
TABLE 1 Gains and Axial Ratios in Broadside Direction for
the Different GNSS Standards
Proposed
Antenna
GNSS Standards
GPS GLONASS Galileo
L
1
L
2
L
5
G
1
G
2
E5a E5b L
1
Gain @zenith
(dBi)
>5.4 >5.8 >3.7 >8.2 >5.2 >3.7 >4.4 >5.4
AR @zenith
(dB)
<1.2 <2.3 <2.1 <1.2 <2 <2.2 <2.3 <1.2
2814 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 12, December 2012 DOI 10.1002/mop
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